James Aspnes

Skip graphs

Skip graphs are a novel distributed data structure, based on skip lists, that provide the full functionality of a balanced tree in a distributed system where resources are stored in separate nodes that may fail at any time. They are designed for use in searching peer-to-peer systems, and by providing the ability to perform queries based on key ordering, they improve on existing search tools that provide only hash table functionality. Unlike skip lists or other tree data structures, skip graphs are highly resilient, tolerating a large fraction of failed nodes without losing connectivity. In addition, simple and straightforward algorithms can be used to construct a skip graph, insert new nodes into it, search it, and detect and repair errors within it introduced due to node failures.

A Theory of Network Localization

In this paper, we provide a theoretical foundation for the problem of network localization in which some nodes know their locations and other nodes determine their locations by measuring the distances to their neighbors. We construct grounded graphs to model network localization and apply graph rigidity theory to test the conditions for unique localizability and to construct uniquely localizable networks. We further study the computational complexity of network localization and investigate a subclass of grounded graphs where localization can be computed efficiently. We conclude with a discussion of localization in sensor networks where the sensors are placed randomly

On-line routing of virtual circuits with applications to load balancing and machine scheduling

In this paper we study the problem of on-line allocation of routes to virtual circuits (both <italic>point-to-point</italic> and <italic>multicast</italic>) where the goal is to route all requests while minimizing the required bandwidth. We concentrate on the case of <italic>Permanent</italic> virtual circuits (i.e., once a circuit is established it exists forever), and describe an algorithm that achieves on <italic>O</italic> (log <italic>n</italic>) competitive ratio with respect to maximum congestin, where <italic>n</italic>is the number of nodes in the network. Informally, our results show that instead of knowing all of the future requests, it is sufficient to increase the bandwidth of the communication links by an <italic>O</italic> (log <italic>n</italic>) factor. We also show that this result is tight, that is, for any on-line algorithm there exists a scenario in which ***(log <italic>n</italic>) increase in bandwidth is necessary in directed networks. We view virtual circuit routing as a generalization of an on-line load balancing problem, defined as follows: jobs arrive on line and each job must be assigned to one of the machines immediately upon arrival. Assigning a job to a machine increases the machines load by an amount that depends both on the job and on the machine. The goal is to minimize the maximum load. For the <italic>related machines</italic> case, we describe the first algorithm that achieves constant competitive ratio. for the <italic>unrelated</italic> case (with <italic>n</italic>machines), we describe a new method that yields <italic>O</italic>(log<italic>n</italic>)-competitive algorithm. This stands in contrast to the natural greed approach, whose competitive ratio is exactly <italic>n</italic>. show that this result is tight, that is, for any on-line algorithm there exists a scenario in which ***(log <italic>n</italic>) increase in bandwidth is necessary in directed networks.

Computation in networks of passively mobile finite-state sensors

We explore the computational power of networks of small resource-limited mobile agents. We define two new models of computation based on pairwise interactions of finite-state agents in populations of finite but unbounded size. With a fairness condition on interactions, we define the concept of stable computation of a function or predicate, and give protocols that stably compute functions in a class including Boolean combinations of threshold-k, parity, majority, and simple arithmetic. We prove that all stably computable predicates are in NL. With uniform random sampling of pairs to interact, we define the model of conjugating automata and show that any counter machine with O(1) counters of capacity O(n) can be simulated with high probability by a protocol in a population of size n. We prove that all predicates computable with high probability in this model are in P ∩ RL. Several open problems and promising future directions are discussed.

Fast randomized consensus using shared memory

Abstract We give a new randomized algorithm for achieving consensus among asynchronous processes that communicate by reading and writing shared registers. The fastest previously known algorithm has exponential expected running time. Our algorithm is polynomial, requiring an expected O ( n 4 ) operations. Applications of this algorithm include the elimination of critical sections from concurrent data structures and the construction of asymptotically unbiased shared coins.

On-line load balancing with applications to machine scheduling and virtual circuit routing

In this paper we study an idealized problem of on-line allocation of routes to virtual circuits where the goal is to minimize the required bandwidth. For the case where virtual circuits continue to exist forever, we describe an algorithm that achieves an O (log n) competitive ratio, where n is the number of nodes in the network. Informally, our results show that instead of knowing all of the future requests, it is sufficient to increase the bandwidth of the communication links by an O(log n) factor. We also show that this result is tight, i.e. for any on-line algorithm there exists a scenario in which O(log n) increase in bandwidth is necessary. We view virtual circuit routing as a generalization of an on-line scheduling problem, and hence a major part of the paper focuses on development of algorithms for non-preemptive on-line scheduling for related and unrelated machines. Specialization of routing to scheduling leads us to concentrate on scheduling in the case where jobs must be assigned immediately upon arrival; assigning a job to a machine increases this machine’s load by an amount that depends both on the job and on the machine. The goal is to minimize the maximum load. For the related machines case, we describe the first algorithm that achieves constant competitive ratio. For the unrekzted case (with n machines), we describe a new method that yields O(log n)-competitive algorithm. This stands in contrast to the natural greedy approach, which we show has only a ~(n) competitive ratio. The virtual circuit routing result follows as a generalization of the unrelated machines case.

On the Computational Complexity of Sensor Network Localization

Determining the positions of the sensor nodes in a network is essential to many network functionalities such as routing, coverage and tracking, and event detection. The localization problem for sensor networks is to reconstruct the positions of all of the sensors in a network, given the distances between all pairs of sensors that are within some radius r of each other. In the past few years, many algorithms for solving the localization problem were proposed, without knowing the computational complexity of the problem. In this paper, we show that no polynomial-time algorithm can solve this problem in the worst case, even for sets of distance pairs for which a unique solution exists, unless RP = NP. We also discuss the consequences of our result and present open problems.

An Introduction to Population Protocols

Population protocols are used as a theoretical model for a collection (or population) of tiny mobile agents that interact with one another to carry out a computation. The agents are identically programmed finite state machines. Input values are initially distributed to the agents, and pairs of agents can exchange state information with other agents when they are close together. The movement pattern of the agents is unpredictable, but subject to some fairness constraints, and computations must eventually converge to the correct output value in any schedule that results from that movement. This framework can be used to model mobile ad hoc networks of tiny devices or collections of molecules undergoing chemical reactions. This chapter surveys results that describe what can be computed in various versions of the population protocol model.

Counting networks

Many fundamental multi-processor coordination problems can be expressed as <italic>counting problems</italic>: Processes must cooperate to assign successive values from a given range, such as addresses in memory or destinations on an interconnection network. Conventional solutions to these problems perform poorly because of synchronization bottlenecks and high memory contention. Motivated by observations on the behavior of sorting networks, we offer a new approach to solving such problems, by introducing <italic>counting networks</italic>, a new class of networks that can be used to count. We give two counting network constructions, one of depth log <italic>n</italic>(1 + log <italic>n</italic>)/2 using <italic>n</italic> log <italic></italic>(1 + log <italic>n</italic>)/4 “gates,” and a second of depth log<supscrpt>2</supscrpt> <italic>n</italic> using <italic>n</italic> log<supscrpt>2</supscrpt> <italic>n</italic>/2 gates. These networks avoid the sequential bottlenecks inherent to earlier solutions and substantially lower the memory contention. Finally, to show that counting networks are not merely mathematical creatures, we provide experimental evidence that they outperform conventional synchronization techniques under a variety of circumstances.

Fault-tolerant routing in peer-to-peer systems

We consider the problem of designing an overlay network and routing mechanism that permits finding resources efficiently in a peer-to-peer system. We argue that many existing approaches to this problem can be modeled as the construction of a random graph embedded in a metric space whose points represent resource identifiers, where the probability of a connection between two nodes depends only on the distance between them in the metric space. We study the performance of a peer-to-peer system where nodes are embedded at grid points in a simple metric space: a one-dimensional real line. We prove upper and lower bounds on the message complexity of locating particular resources in such a system, under a variety of assumptions about failures of either nodes or the connections between them. Our lower bounds in particular show that the use of inverse power-law distributions in routing, as suggested by Kleinberg [5], is close to optimal. We also give heuristics to efficiently maintain a network supporting efficient routing as nodes enter and leave the system. Finally, we give some experimental results that suggest promising directions for future work.